A popular model in time series analysis is the autoregression model. It explains the current value of a modeled process by means of a weighted average of its past values. We focus on the case where the modeled variable stands for counts, i.e., it takes values in the set of nonnegative integers. We also focus on online modeling, where the arriving observations sequentially update estimates of the parameters. The emphasis is put on low computational requirements, opening the way towards high-rate real world applications. This contribution describes our initial results in this domain. The solution is based on a Poisson autoregression model, where the linear predictor and the modeled variable are canonically linked by the logarithmic function. The adopted Bayesian estimation framework relies on an analytical approximation of the Poisson model by a Gaussian density. A Gaussian prior then provides an analytically tractable posterior estimator. Two examples demonstrate the feasibility of the solution. We demonstrate a potential of our approach on the simulated data as well as COVID-19 data and notice advantageous performance of considered methods especially for the use in machine learning.